A critical evaluation of dynamical systems models of bipolar disorder

Bipolar disorder (BD) is a mood disorder involving recurring (hypo)manic and depressive episodes. The inherently temporal nature of BD has inspired its conceptualization using dynamical systems theory, which is a mathematical framework for understanding systems that evolve over time. In this paper, we provide a critical review of the dynamical systems models of BD. Owing to the heterogeneity of methodological and experimental designs in computational modeling, we designed a structured approach that parallels the appraisal of animal models by their face, predictive, and construct validity. This tool, the validity appraisal guide for computational models (VAG-CM), is not an absolute measure of validity, but rather a guide for a more objective appraisal of models in this review. We identified 26 studies published before November 18, 2021 that proposed generative dynamical systems models of time-varying signals in BD. Two raters independently applied the VAG-CM to the included studies, obtaining a mean Cohen’s κ of 0.55 (95% CI [0.45, 0.64]) prior to establishing consensus ratings. Consensus VAG-CM ratings revealed three model/study clusters: data-driven models with face validity, theory-driven models with predictive validity, and theory-driven models lacking all forms of validity. We conclude that future modeling studies should employ a hybrid approach that first operationalizes BD features of interest using empirical data to achieve face validity, followed by explanations of those features using generative models with components that are homologous to physiological or psychological systems involved in BD, to achieve construct validity. Such models would be best developed alongside long-term prospective cohort studies involving a collection of multimodal time-series data. We also encourage future studies to extend, modify, and evaluate the VAG-CM approach for a wider breadth of computational modeling studies and psychiatric disorders.


OVERVIEW OF DYNAMICAL SYSTEMS
In order to facilitate interpretation of the models uncovered by our review, this section first introduces some basic technical aspects of dynamical systems using simple examples. A dynamical system is a system whose state evolves over time in accordance to an "evolution rule", which can be mathematically described by sets of differential equations in the continuous setting, or difference equations in the discrete setting. Here, we provide a basic introduction to models based on differential equations. Our first model is demonstrative of continuous dynamical systems that reach a stable equilibrium (a stable state or fixed point). These models may have one or more stable states, but are characterized by the system converging to, and remaining at, these points or limit sets until perturbed. The specific stable state to which a system arrives may depend on several factors, including the starting point. Example 1 describes a simple dynamical system modelling growth of a population under (A) space constraints and (B) predation, in which the population size reaches a stable equilibrium.
Example 1 (A System with Stable States). A classical dynamical system describes the growth of a population whose size at time 0 ≤ t is denoted x(t) or simply x for parsimony. The rate of change in x, denotedẋ (alternatively x ′ (t) or dx/dt) isẋ which is known as the logistic equation. Here, coefficient x max denotes the maximum possible population size, and a is a scalar value identifying the growth rate. If a > 1, then the population will grow over time, whereas it will shrink if a < 1. The term (1 − x/x max ) indicates the proportion of population capacity still available for reproduction. As x approaches x max , the value of (1 − x/x max ) approaches 0 and reproduction stops. Some populations are also subject to predation. This can be incorporated into Equation 1 by adding a predation function bx 2 /(c 2 + x 2 ), where b > 0 is a constant representing the rate of predation, and c > 0 represents the size of the population at which 50% of the maximum predation rate is reached (note that this is equivalent to the biochemical Hill function with exponent n = 2). Equation 1 thus becomeṡ (2) Supplemental Figure 1 is a stream plot showing the trajectories of population size (as modelled by Equation 2) across time. For initial population sizes greater than 0, the population will converge on a stable equilibrium value. The rate at which the population approaches that fixed point is dependent on the initial population size. Supplemental Figure 2: The van der Pol oscillator (VPO) [8,9] across values of the nonlinearity/damping parameter µ. Each row of plots illustrates a VPO for different values of µ. The left and right columns show the phase space portraits, and corresponding time series, respectively.
Instead of demonstrating stable fixed points, some systems show solutions that converge to limit cycle attractors, resulting in periodic oscillations. One such class of systems that will be encountered across many studies in our review [1][2][3][4][5][6][7] is the relaxation oscillator (RO), most notably the van der Pol oscillator (VPO) [8,9]. Although population dynamics models, including those adapted for analysis of biochemical kinetics can also produce oscillatory behaviour [10,11], Example 2 will introduce the VPO. [8,9] is a one-dimensional system with state x(t) (here simply x) is defined according to the following second order ordinary differential equation:ẍ

Example 2 (A System with Periodic Oscillations). A van der Pol oscillator
where "second order" refers to the fact that these dynamics are expressed in terms of the second derivative of x with respect to time:ẍ = x ′′ (t) = d 2 x/dt 2 . The parameter µ is a scalar value representing the amount of damping or nonlinearity. Supplemental Figure 2 shows the VPO behaviour across different values of µ. When µ < 0, the system will show a progressive damping of oscillations until x approaches 0. When µ = 0, the system shows simple harmonic motion. When µ > 0, we observe limit cycles (stable periodic trajectories to which the system will converge from multiple different starting points). Supplemental Figure 2 also demonstrates the phenomenon of a "relaxation oscillation," whereby the system alternates between slow and rapid movements (depicted here when µ = 7).
One can appreciate why RO models might be proposed for BD mood dynamics. Superficially, BD consists of episodes in which an individual persists for some time, followed ostensibly by transition into a Rather, naturally oscillating systems are often characterized by behaviour that appears almost random. Yet, seemingly random behaviour can emerge from a totally deterministic system in a phenomenon known as chaotic dynamics.

Example 3 (A Chaotic Continuous Dynamical System).
Huber et al. [1] sought to model episodes of mania and depression as discrete (on/off) events, since their primary concern was the rhythmicity of episodes and inter-episode timing. This was implemented using a model of neuronal spiking illustrated graphically in Supplemental Figure 3, and characterized by the following ordinary differential equation, where x is analogous to a neuronal membrane potential, τ x is a relaxation time constant, S is a constant external input, and W is Gaussian noise. The set A = {he, hi, le, li} identifies different excitatory and inhibitory elements of the system. That is, the system contains high-threshold excitatory and inhibitory elements A he and A hi , as well as low-threshold excitatory and inhibitory elements A le and A li , respectively. These are coupled to the state variable x according to "synaptic weights" w i ∈ {w he , w hi , w le , w li } and activation threshold constants Huber et al. [1] set the exponent v to 2 for A he , and to 1 otherwise. The system simulated by Huber et al. [1] can be conceptualized as a mixture of excitatory/inhibitory oscillating subsystems, whose states A i ∈ {A he , A hi , A le , A li } themselves evolve over time according to the following differential equation: Here, τ i is a relaxation time constant, and is a sigmoid with slope β i and half-activation constant x i,0.5 . At different levels of deterministic input (S), this system is capable of simulating fixed-point or steady state dynamics, as well as periodic oscillations, limit cycles, and chaotic behaviour. Supplemental Figure 4 demonstrates this behaviour. With S ≈ 10 or below, the system undergoes dampened oscillations to a limiting fixed point. At S ≈ 50, we observe regular periodic spiking, and periodic bursts at S ≈ 100. At levels of S between approximately 175 and 325, we observe chaotic inter-spike intervals (see Huber et al. [1] for further detail through bifurcation plots).
In this section, we have introduced simple examples of continuous dynamical systems capable of generating multiple forms of behaviour, including fixed-points, stable oscillations and limit cycles, as well as chaotic dynamics. Two of these models, the RO (Example 2) and neuronal model (Example 3), are directly implemented in studies that will be discussed in the Results and Discussion of the main body. □ The intervention resulting in transition between models is explicitly defined □ Transformation of one model into the other corresponds to transformation between the target states OR Transformation of one model into the other predicts transformation between target states that can be verified empirically □ Degree to which transformation between models corresponds to transformation between target states is quantified statistically, and deemed to be the best fit, at least in comparison to a "null" or random transformation OR Transformation of one model into the other predicts a transformation that should be observed in between the target states/conditions in the real-world. Prediction is quantitative, empirically verifiable, and includes degree of uncertainty

Subscale 3: Construct Validity
Definition: The degree of homology between the model and mechanisms that are empirically or theoretically deemed to underlie features of the target condition of interest.
1. There is a real and identifiable or plausible mechanism underlying the target condition/state □ Explicitly identified □ Empirically or theoretically shown to be involved/associated with target features being explained OR Model makes explicit testable predictions about the existence of this underlying mechanism □ The model architecture is homologous to the mechanism of interest, at an appropriate level of abstraction 2. Model components are defined and linked to specific components of underlying mechanism □ Functional arrangement/connections between model components defined and linked to underlying mechanisms □ Multiple models were evaluated with competing mechanisms

INTER-RATER RELIABILITY RESULTS
Cohen's κ for each individual paper reviewed is plotted in Supplemental Figure 5.

The model aims to describe a real-world phenomenon (i.e. a target state/condition vis a vis comparators)
Target condition defined clearly Degree to which transformation between models corresponds to transformation between target states is quantified statistically, and deemed to be the best fit, at least in comparison to a "null" or random transformation OR Transformation of one model into the other predicts a transformation that should be observed in between the target states/conditions in the real-world. Prediction is quantitative, empirically verifiable, and includes degree of uncertainty